A property of Dunford-Pettis type in topological groups

Martín-Peinador, E.; Tarieladze, V.

doi:10.1090/S0002-9939-03-07249-6

A property of Dunford-Pettis type in topological groups
HTML articles powered by AMS MathViewer

by E. Martín-Peinador and V. Tarieladze PDF

Proc. Amer. Math. Soc. 132 (2004), 1827-1837 Request permission

Abstract:

The property of Dunford-Pettis for a locally convex space was introduced by Grothendieck in 1953. Since then it has been intensively studied, with especial emphasis in the framework of Banach space theory. In this paper we define the Bohr sequential continuity property (BSCP) for a topological Abelian group. This notion could be the analogue to the Dunford-Pettis property in the context of groups. We have picked this name because the Bohr topology of the group and of the dual group plays an important role in the definition. We relate the BSCP with the Schur property, which also admits a natural formulation for Abelian topological groups, and we prove that they are equivalent within the class of separable metrizable locally quasi-convex groups. For Banach spaces (or for metrizable locally convex spaces), considered in their additive structure, we show that the BSCP lies between the Schur and the Dunford-Pettis properties.

References

L. Aussenhofer, Contributions to the duality theory of abelian topological groups and to the theory of nuclear groups, Dissertationes Math. (Rozprawy Mat.) 384 (1999), 113. MR 1736984
Wojciech Banaszczyk, Additive subgroups of topological vector spaces, Lecture Notes in Mathematics, vol. 1466, Springer-Verlag, Berlin, 1991. MR 1119302, DOI 10.1007/BFb0089147
W. Banaszczyk and E. Martín-Peinador, The Glicksberg theorem on weakly compact sets for nuclear groups, Papers on general topology and applications (Amsterdam, 1994) Ann. New York Acad. Sci., vol. 788, New York Acad. Sci., New York, 1996, pp. 34–39. MR 1460814, DOI 10.1111/j.1749-6632.1996.tb36794.x
Fernando Bombal and Ignacio Villanueva, On the Dunford-Pettis property of the tensor product of $C(K)$ spaces, Proc. Amer. Math. Soc. 129 (2001), no. 5, 1359–1363. MR 1712870, DOI 10.1090/S0002-9939-00-05662-8
Bruguera, M., Grupos topológicos y grupos de convergencia: estudio de la dualidad de Pontryagin, Doctoral Dissertation, 1999.
B. Cascales and J. Orihuela, On compactness in locally convex spaces, Math. Z. 195 (1987), no. 3, 365–381. MR 895307, DOI 10.1007/BF01161762
Pilar Cembranos, The hereditary Dunford-Pettis property on $C(K,E)$, Illinois J. Math. 31 (1987), no. 3, 365–373. MR 892175
Pilar Cembranos, The hereditary Dunford-Pettis property for $l_1(E)$, Proc. Amer. Math. Soc. 108 (1990), no. 4, 947–950. MR 1004415, DOI 10.1090/S0002-9939-1990-1004415-6
M. J. Chasco, Pontryagin duality for metrizable groups, Arch. Math. (Basel) 70 (1998), no. 1, 22–28. MR 1487450, DOI 10.1007/s000130050160
M. J. Chasco, E. Martín-Peinador, and V. Tarieladze, On Mackey topology for groups, Studia Math. 132 (1999), no. 3, 257–284. MR 1669662, DOI 10.4064/sm-132-3-257-284
W. W. Comfort, Salvador Hernández, and F. Javier Trigos-Arrieta, Relating a locally compact abelian group to its Bohr compactification, Adv. Math. 120 (1996), no. 2, 322–344. MR 1397086, DOI 10.1006/aima.1996.0042
W. W. Comfort and K. A. Ross, Topologies induced by groups of characters, Fund. Math. 55 (1964), 283–291. MR 169940, DOI 10.4064/fm-55-3-283-291
Joe Diestel, A survey of results related to the Dunford-Pettis property, Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces (Univ. North Carolina, Chapel Hill, N.C., 1979) Contemp. Math., vol. 2, Amer. Math. Soc., Providence, R.I., 1980, pp. 15–60. MR 621850
Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
Salvador Hernández, Jorge Galindo, and Sergio Macario, A characterization of the Schur property by means of the Bohr topology, Topology Appl. 97 (1999), no. 1-2, 99–108. Special issue in honor of W. W. Comfort (Curacao, 1996). MR 1676674, DOI 10.1016/S0166-8641(98)00071-6
Irving Glicksberg, Uniform boundedness for groups, Canadian J. Math. 14 (1962), 269–276. MR 155923, DOI 10.4153/CJM-1962-017-3
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der mathematischen Wissenschaften, Band 115, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156915
Elena Martín Peinador, A reflexive admissible topological group must be locally compact, Proc. Amer. Math. Soc. 123 (1995), no. 11, 3563–3566. MR 1301516, DOI 10.1090/S0002-9939-1995-1301516-4
Dieter Remus and F. Javier Trigos-Arrieta, Abelian groups which satisfy Pontryagin duality need not respect compactness, Proc. Amer. Math. Soc. 117 (1993), no. 4, 1195–1200. MR 1132422, DOI 10.1090/S0002-9939-1993-1132422-4
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
N. Th. Varopoulos, Studies in harmonic analysis, Proc. Cambridge Philos. Soc. 60 (1964), 465–516. MR 163985, DOI 10.1017/s030500410003797x